L-functions form an integral part of the 'L-functions and Modular Forms Database' with far-reaching implications. In perspective, Riemann zeta function is the simplest example of an L-function. Riemann hypothesis refers to the 1859 proposal by Bernhard Riemann whereby all nontrivial zeros are [mathematically] conjectured to lie on the critical line of this function. This proposal is equivalently stated in this research paper as all nontrivial zeros are [geometrically] conjectured to exactly match the 'Origin' intercepts of this function. Deeply entrenched in number theory, prime number theorem entails analysis of prime counting function for prime numbers. Solving Riemann hypothesis would enable complete delineation of this important theorem. Involving proposals on the magnitude of prime gaps and their associated sets of prime numbers, Twin prime conjecture deals with prime gap = 2 (representing twin primes) and is thus a subset of Polignac's conjecture which deals with all even number prime gaps = 2, 4, 6,... (representing prime numbers in totality except for the first prime number '2'). Both nontrivial zeros and prime numbers are Incompletely Predictable entities allowing us to employ our novel Virtual Container Research Method to solve the associated hypothesis and conjectures.
Category:Number Theory

Unknown Algorithms for Finding Prime Numbers Among Odd Numbers

The discovered algorithm for extracting prime numbers from the natural series is alternative to both the Eratosthenes lattice and Sundaram and Atkin's sentences. The distribution of prime numbers does not have a formula, but if the number is one less than the prime number is an exponent of the integers, then there are no two scalar scalars whose sum is equal to the third integer in the same degree. This is the sound of P. Fermat's Great Theorem, the proof of which he could begin by using the Minor theorem known to him. The first part of the proof is here restored. But how did P. Fermat finish it?
Category:Number Theory

A Recreative Method to Obtain from a Given Prime Larger Primes Based on the Powers of 3

In this paper I present a method to obtain from a given prime p1 larger primes, namely inserting before of a digit of p1 a power of 3, and, once a prime p2 is obtained, repeating the operation on p2 and so on. By this method I obtained from a prime with 9 digits a prime with 36 digits (the steps are showed in this paper) using just the numbers 3, 9(3^2), 27(3^3) and 243(3^5).
Category:Number Theory

Open Number Theory Problem

In math the the 7 Clay Math unsolved problems? Another problem is the question if there is a God(s)? In my paper the purpose is to explain that in the end we all meet our maker and that man does not have the power to cheat death. Like the Riemann Zeta function that remains unsolved and when solved will give insight to distribution of the Primes, giving or solving this open-end problem will help me solve a problem. This is the only problem I have not been able to solve and I am open sourcing it.
Category:Number Theory

Distribution of the Residues and Cycle Counting

In this paper we take a closer look to the distribution of the residues of squarefree natural numbers and explain an algorithm to compute those distributions.
We also give some conjectures about the minimal number of cycles in the squarefree arithmetic progression and explain an algorithm to compute this minimal numbers.
Category:Number Theory

Analytic Demonstrations on the Fourfold Root Topics of Primes

This paper is concerned with formulation and demonstration of new versions of equations that can help us resolve problems concerning maximal gaps between consecutive prime numbers, the number of prime numbers at a given magnitude and the location of nth prime number. There is also a mathematical argument on why prime numbers as elementary identities on their own respect behave the way they do. Given that the equations have already been formulated, there are worked out examples on numbers that represent different cohorts. This paper has therefore attempted to formulate an equation that approximates the number of prime numbers at a given magnitude, from N=3 to N=〖10〗^25. Concerning the location of an nth prime number, the paper has devised a method that can help us locate a given prime number within specified bounds. Nonetheless, the paper has formulated an equation that can help us determine extremely bounded gaps. Lastly, using trans-algebraic number theory method, the paper has shown that unpredictable behaviors of prime numbers are due to their identity nature.
Category:Number Theory

Conjecture on the Pairs of Primes Obtained Inserting N with Digit Sum 12 After the First Digit of Twin Primes

In this paper I conjecture that for any pair of twin primes [p, q], p ≥ 11, there exist a number n having the sum of its digits equal to 12 such that inserting n after the first digit of p respectively q are obtained two primes (almost always twins, as in the case [1481, 1483] where n = 48 is inserted in [11, 13], beside the case that the first digit of twins is different, as in the case [5669, 6661] where n = 66 is inserted in [59, 61]).
Category:Number Theory

Primes Obtained Concatenating a Poulet Number P with (s-1)÷n Where S Digits Sum of P and N is 2, 3 or 6

In this paper I conjecture that there exist an infinity of Poulet numbers P such that concatenating P to the left with the number (s(P) – 1)/2, where s is the sum of digits of P, is obtained a prime; also I make the same conjecture for (s(P) – 1)/3 respectively for (s(P) – 1)/6.
Category:Number Theory

Primes Obtained Concatenating to the Left a Prime Having an Odd Prime Digit Sum S with a Divisor of S-1

In a previous paper, “Primes obtained concatenating a Poulet number P with (s - 1)/n where s digits sum of P and n is 2, 3 or 6”, I noticed that in almost all the cases that I considered if a prime was obtained through this concatenation than the digits sum of P was a prime. That gave me the idea for this paper where I observe that for many primes p having an odd prime digit sum s there exist a prime obtained concatenating p to the left with a divisor of s – 1 (including 1 and s – 1).
Category:Number Theory

Primes Obtained Concatenating to the Left a Prime Having an Odd Prime Digit Sum S with a Multiple of S-1

In a previous paper, “Primes obtained concatenating to the left a prime having an odd prime digit sum s with a divisor of s - 1”, I observed that for many primes p having an odd prime digit sum s there exist a prime obtained concatenating p to the left with a divisor of s – 1. In this paper I conjecture that for any prime p, p ≠ 5, having an odd prime digit sum s there exist an infinity of primes obtained concatenating to the left p with multiples of s – 1. Yet I conjecture that there exist at least a prime obtained concatenating n*(s – 1) with p such that n < sqr s.
Category:Number Theory

The Wave Function ψ of the Riemann Zeta Function ζ(0.5+it):

There is exciting research trying to connect the nontrivial zeros of the Riemann Zeta function to Quantum mechanics as a breakthrough towards proving the 160-year-old Riemann Hypothesis. This research offers a radically new approach.
Most research up to this point have focused only on mapping the nontrivial zeros directly to eigenvalues. Those attempts have failed or didn’t yield any new breakthrough. This research takes a radically different approach by focusing on the quantum mechanical properties of the wave graph of Zeta as ζ(0.5+it) and not the nontrivial zeros directly. The conjecture is that the wave forms in the graph of the Riemann Zeta function ζ(0.5+it) is a wave function ψ. It is made of a Complex version of the Parity Operator wave function. The Riemann Zeta function consists of linked Even and Odd Parity Operator wave functions on the critical line. From this new approach, it shows the Complex version of the Parity Operator wave function is Hermitian and it eigenvalues matches the zeros of the Zeta function.
Category:Number Theory

A Criterion Arising from Explorations Pertaining to the Oesterle-Masser Conjecture

Using an extension of the idea of the radical of a number, as well as a few other ideas, it is indicated as to why one might expect the Oesterle-Masser conjecture to be true. Based on structural elements arising from this proof, a criterion is then developed and shown to be potentially sufficient to resolve two relatively deep conjectures about the structure of the prime numbers. A sketch is consequently provided as to how it might be possible to demonstrate this criterion, borrowing ideas from information theory and cybernetics.
Category:Number Theory